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In this paper we mainly consider the univalence problem of $F=P_f$, where $f$ belongs to some subclasses of ${\\mathcal S}$. Among several sharp results and non-sharp results, we also show that if $f\\in {\\mathcal S}$, then $F \\in {\\mathcal U}$ in the disk $|z|<r$ with $r\\leq r_6\\approx 0.360794$ and conjecture that th"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1503.04931","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-03-17T06:22:24Z","cross_cats_sorted":[],"title_canon_sha256":"9c22f00b24d5f80ad04612e2ca075a49b58363955cebd5ba368fd6130ee4d460","abstract_canon_sha256":"9bc85b3764cb27cb3612a3bf4a0d5233cb67c06f49ddb92e461cacb95934a967"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:22:21.082275Z","signature_b64":"aylMSVDNbZUFoLqC8ag8w6A2XCfLIe7FU0KGlPl5sl4bSbZMZG8xAsL/1KgMQD+7oKX5EqxZPw8EeNq4C847DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2296bccdc8c84aa32f3895e7aee79e95bf013ba2801457cba550c58fefd06cb2","last_reissued_at":"2026-05-18T02:22:21.081562Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:22:21.081562Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Where is $f(z)/f'(z)$ univalent?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Karl-Joachim Wirths, Milutin Obradovi\\'c, Saminathan Ponnusamy","submitted_at":"2015-03-17T06:22:24Z","abstract_excerpt":"Let ${\\mathcal S}$ denote the family of all univalent functions $f$ in the unit disk $\\ID$ with the normalization $f(0)=0= f'(0)-1$. 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